Modelling the pressure in the Earth's interior is a common problem in Earth sciences. In this study we propose a method based on the conservation of the momentum of a fluid by using a hydrostatic scenario or a uniformly moving fluid to approximate the pressure. This results in a partial differential equation (PDE) that can be solved using classical numerical methods. In hydrostatic cases, the computed pressure is the lithostatic pressure. In non-hydrostatic cases, we show that this PDE-based approach better approximates the total pressure than the classical 1D depth-integrated approach. To illustrate the performance of this PDE-based formulation we present several hydrostatic and non-hydrostatic 2D models in which we compute the lithostatic pressure or an approximation of the total pressure, respectively. Moreover, we also present a 3D rift model that uses that approximated pressure as a time-dependent boundary condition to simulate far-field normal stresses. This model shows a high degree of non-cylindrical deformation, resulting from the stress boundary condition, that is accommodated by strike-slip shear zones. We compare the result of this numerical model with a traditional rift model employing free-slip boundary conditions to demonstrate the first-order implications of considering “open” boundary conditions in 3D thermo-mechanical rift models.

In Earth sciences and geodynamic modelling, computing the pressure can be essential.
Specifically, numerous regional thermo-mechanical studies use the lithostatic pressure or a reference pressure based on some density structure as a normal stress boundary condition

The common approach to compute a reference pressure

Although evaluating Eq. (

a mesh with cell edges (2D) or faces (3D) that are not aligned with the gravity vector (Fig.

an unstructured mesh (Fig.

a density structure (or gravity vector) that is spatially varying,

a parallel decomposition of the mesh (Fig.

time dependence in the density or mesh coordinates that requires continual re-evaluation of the reference pressure.

Schematic representation of meshes for which computing an integral in the vertical direction can be challenging:

To compute

Moreover, when the density structure evolves with time as deformation occurs, the pressure gradients may no longer be aligned with the gravitational acceleration vector. In these non-hydrostatic cases, this pressure is not lithostatic. However, to be able to provide an approximation for the total pressure or to use stress boundary conditions, it is still important to approximate the total pressure in the best possible way.

For these reasons, we propose an efficient mesh and numerical method (finite elements, finite differences, finite volumes, etc.) to compute a reference pressure associated with the density structure of a domain in hydrostatic cases or an approximation of the total pressure for non-hydrostatic cases for all scenarios given above by solving a partial differential equation (PDE) derived from the conservation of the non-inertial momentum equation for an incompressible fluid. We also present thermo-mechanical numerical models and static numerical models applied to Earth sciences and geodynamics to show the usefulness of this approach.

For an incompressible fluid in a domain

To define the pressure associated with the density structure we make the “ansatz” that

As such, there is no unique solution to Eq. (

Equation (

Assuming that

A unique solution to Eq. (

Along the surface of the domain, which represents the free surface of the Earth, we impose

Two different boundary conditions are introduced to constrain

The first constraint states that

Since the unit vector

Equations (

We note for domains with boundaries parallel to either

One may also consider employing both Eqs. (

Nevertheless, for an arbitrarily shaped domain, using boundary conditions (

To define the weak formulation of the PPE we will use functions that are square integrable in the sense of Lebesgue, i.e.

Splitting the surface integrals over the two segments

From Eq. (

The strong (Eq.

In this section we provide several numerical models to show the following items:

the accuracy and consistency of the method for hydrostatic cases,

the accuracy of the approximation of the total pressure in non-hydrostatic cases,

the effect of using the depth-integrated approach and the pressure Poisson problem approach to impose boundary conditions on the momentum equation,

the usefulness of the method for 3D geodynamic thermo-mechanical modelling.

To compare the numerical solution of Eq. (

Pressure for non-dimensioned hydrostatic cases. Panels

Pressure for a hydrostatic case in a half annulus approximating a simplified and idealized layered Earth.

We define the domain

This case assumes a constant density,

This case assumes a continuous depth-varying density

This case assumes a discontinuous density such that

A finite-element (FE) method employing an unstructured triangular mesh with a

The 2D half-annulus model aims to show the efficiency of the method when computing the lithostatic pressure
in a body with a radial gravity vector and concentric density structure (Fig.

The numerical solution extracted at

The four hydrostatic models clearly illustrate that the solutions obtained using the depth-integrated approach and the pressure Poisson equation (with one set of boundary constraints) are equivalent for scenarios that admit a hydrostatic solution.

Here, we show the differences and accuracy of the depth-integrated equation (Eq.

Non-dimensional “global” model for a large domain with a topographic perturbation.

Non-dimensional “regional” model for a small domain with a varying topography.

We define a large domain

We solve the flow problem described by Eqs. (

The difference between the total pressure and the approximated pressure

Nevertheless, modelling a domain 20 times larger than the domain of interest can hardly be achieved in practice, mainly due to the numerical cost it represents.
Thus, the boundary conditions are a first-order component of regional models in order to best capture the global behaviour and interactions in a region without having to model the whole Earth.
Therefore, we define a smaller domain

In this case we aim to apply a normal stress on the boundaries of our regional model that will generate a flow similar (or close) to the flow generated in the global model.
To achieve this we present two models using Eqs. (

Figure

In Fig.

Figure

Curves showing the pressure from the regional and global models along the profiles

These simple tests demonstrate that the approximated pressure

The boundary conditions used to solve the pressure Poisson problem are stated in Eqs. (

Pressure field

In Fig.

Moreover, Fig.

Pressure field

As previously noted, discretizations employing the weak form with Eq. (

To simulate the long-term evolution of the deformation of the lithosphere, we solve the stationary, non-inertial form of the conservation of momentum described by Eq. (

The numerical solution of Eqs. (

To model the strain localization we use non-linear visco-plastic rheologies expressed in term of viscosity. The ductile parts of the domain are simulated using an Arrhenius flow law for dislocation creep

The modelled domain contains four initial flat layers representing the upper continental crust, the lower continental crust, the lithosphere mantle, and the asthenosphere mantle, respectively (Fig.

The initial density distribution follows the lithologies and is reported in Table

Moreover, the initial temperature field is computed as a steady-state solution of the heat equation

Physical parameters for the thermo-mechanical rift model.

To show the influence of the normal stress boundary condition, we compare two rift models. In the reference model, an extension velocity of

The second rift model (Fig.

To account for the density evolution through time due to the deformation and material advection, Eq. (

The bottom of the domain is prescribed as an inflow condition balancing the outflow, and the surface of the domain is a free surface where the mesh deforms according to the computed velocity field. These Neumann boundary conditions allow material to flow both in and out through the boundary depending only on the Dirichlet boundary conditions and deformation that occurs inside the modelled domain.

In the context of our finite-element forward model, we also solve the pressure Poisson problem using finite elements.
As such, to compute

As a demonstration of the computed

The model using free-slip boundary conditions displays a cylindrical deformation pattern that could be reduced to a two-dimensional model. As shown by the shear zone orientation and strain regime, the deformation is only extensional and perpendicular to the extension direction (Figs.

Map view of the strain regime evolution in time and space of the model with

Map view of the models with free-slip (left) and normal stress (right) boundary conditions. The dashed lines in the upper panels indicated by

In contrast, the model using the

Map view of the

Recall that the starting point of defining the PPE was purely algebraic, with the sole intention of
removing the non-uniqueness associated with Eq. (

Rather than enforcing Eq. (

“Inflow” in the context of Eq. (

Compared to the PPE, the hyperbolic formulation has several disadvantages.

We have less freedom to specify how

The formulation may place restrictions on the shape of

The lack of flexibility in controlling the boundary behaviour of

The spatial discretization required for the accurate solution of Eq. (

Pressure in non-dimensional models with constant density

From a linear algebra perspective, the non-uniqueness of Eq. (

Since the PPE is a second-order PDE, the formulation permits a range of
possible boundary constraints on

Experiments showed that the PPE approach better approximates the total pressure computed from the momentum equation.
Therefore, its use as a boundary condition (or as an initial guess) for the pressure field to solve the momentum equation
is preferred over hydrostatic solutions associated with Eq. (

In the geodynamic rift model, using the pressure computed with Eq. (

Two approaches can be considered to compute the pressure using Eq. (

In this study we presented a method to compute a reference pressure associated with the density structure of a domain in which we cast the problem in terms of a partial differential equation (PDE). From a practical standpoint, the PDE approach is generic (it is applicable to all spatial discretization and on any type of computational grid), efficient, and applicable in parallel computing environments. From the modelling perspective, the PDE approach has specific advantages, for example in models with a variable density structure (stationary or time-dependent) and models that employ a reference pressure as a boundary condition of the flow problem (stationary or time-dependent problems). Re-evaluating that pressure in time-dependent problems is not problematic (even if the mesh deforms) since solving the Poisson problem can be performed using optimal preconditioners (e.g. geometric or algebraic multigrid preconditioners). Importantly, the time to solve the pressure Poisson problem is a small fraction of the time required to solve the linear (or non-linear) incompressible viscous flow problem. Moreover, we also demonstrate that the PDE formulation results in a better approximation of the total pressure than the 1D depth-integrated approach in non-hydrostatic cases.

Lastly, we showed in the context of 3D geodynamic models of continental rifting that using a reference pressure as a boundary condition within the flow problem resulted in non-cylindrical velocity fields. These 3D velocity fields produced strain localization in the lithosphere along large-scale strike-slip shear zones and the formation and evolution of triple junctions.

The code

The supplement related to this article is available online at:

AJ and DAM conceptualized the project and made the mathematical and software development. AJ and DAM conceived, designed, and performed the experiments. AJ analysed the results. AJ and DAM drafted and corrected the manuscript. DAM procured funding and supervised the project.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (

This project was supported by NSF Award EAR-2121666. Anthony Jourdon has been supported by the European Union's Horizon 2020 research and innovation programme (TEAR ERC Starting, grant no. 852992).

This paper was edited by Taras Gerya and reviewed by Rene Gassmoeller and Cedric Thieulot.